Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1360,2,Mod(1089,1360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1360, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1360.1089");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1360 = 2^{4} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1360.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | no |
| Analytic conductor: | \(10.8596546749\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 170) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 5\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(341\) | \(511\) | \(817\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1089.1 |
|
0 | − | 3.14134i | 0 | 1.38900 | − | 1.75233i | 0 | − | 3.50466i | 0 | −6.86799 | 0 | ||||||||||||||||||||||||||||||||
| 1089.2 | 0 | − | 2.62620i | 0 | −2.19388 | − | 0.432320i | 0 | − | 0.864641i | 0 | −3.89692 | 0 | |||||||||||||||||||||||||||||||||
| 1089.3 | 0 | − | 0.484862i | 0 | 1.80487 | + | 1.32001i | 0 | 2.64002i | 0 | 2.76491 | 0 | ||||||||||||||||||||||||||||||||||
| 1089.4 | 0 | 0.484862i | 0 | 1.80487 | − | 1.32001i | 0 | − | 2.64002i | 0 | 2.76491 | 0 | ||||||||||||||||||||||||||||||||||
| 1089.5 | 0 | 2.62620i | 0 | −2.19388 | + | 0.432320i | 0 | 0.864641i | 0 | −3.89692 | 0 | |||||||||||||||||||||||||||||||||||
| 1089.6 | 0 | 3.14134i | 0 | 1.38900 | + | 1.75233i | 0 | 3.50466i | 0 | −6.86799 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1360.2.e.c | 6 | |
| 4.b | odd | 2 | 1 | 170.2.c.b | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 1360.2.e.c | 6 | |
| 5.c | odd | 4 | 1 | 6800.2.a.bk | 3 | ||
| 5.c | odd | 4 | 1 | 6800.2.a.bp | 3 | ||
| 12.b | even | 2 | 1 | 1530.2.d.g | 6 | ||
| 20.d | odd | 2 | 1 | 170.2.c.b | ✓ | 6 | |
| 20.e | even | 4 | 1 | 850.2.a.p | 3 | ||
| 20.e | even | 4 | 1 | 850.2.a.q | 3 | ||
| 60.h | even | 2 | 1 | 1530.2.d.g | 6 | ||
| 60.l | odd | 4 | 1 | 7650.2.a.dj | 3 | ||
| 60.l | odd | 4 | 1 | 7650.2.a.do | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.2.c.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 170.2.c.b | ✓ | 6 | 20.d | odd | 2 | 1 | |
| 850.2.a.p | 3 | 20.e | even | 4 | 1 | ||
| 850.2.a.q | 3 | 20.e | even | 4 | 1 | ||
| 1360.2.e.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 1360.2.e.c | 6 | 5.b | even | 2 | 1 | inner | |
| 1530.2.d.g | 6 | 12.b | even | 2 | 1 | ||
| 1530.2.d.g | 6 | 60.h | even | 2 | 1 | ||
| 6800.2.a.bk | 3 | 5.c | odd | 4 | 1 | ||
| 6800.2.a.bp | 3 | 5.c | odd | 4 | 1 | ||
| 7650.2.a.dj | 3 | 60.l | odd | 4 | 1 | ||
| 7650.2.a.do | 3 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1360, [\chi])\):
|
\( T_{3}^{6} + 17T_{3}^{4} + 72T_{3}^{2} + 16 \)
|
|
\( T_{7}^{6} + 20T_{7}^{4} + 100T_{7}^{2} + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 17 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
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| $7$ |
\( T^{6} + 20 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T + 2)^{6} \)
|
| $13$ |
\( T^{6} + 17 T^{4} + \cdots + 16 \)
|
| $17$ |
\( (T^{2} + 1)^{3} \)
|
| $19$ |
\( (T^{3} - T^{2} - 24 T - 20)^{2} \)
|
| $23$ |
\( T^{6} + 68 T^{4} + \cdots + 1024 \)
|
| $29$ |
\( (T^{3} + 17 T^{2} + \cdots - 50)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} - 46 T + 74)^{2} \)
|
| $37$ |
\( T^{6} + 308 T^{4} + \cdots + 1032256 \)
|
| $41$ |
\( (T^{3} - 16 T^{2} + \cdots + 16)^{2} \)
|
| $43$ |
\( T^{6} + 72 T^{4} + \cdots + 1024 \)
|
| $47$ |
\( T^{6} + 257 T^{4} + \cdots + 440896 \)
|
| $53$ |
\( T^{6} + 265 T^{4} + \cdots + 583696 \)
|
| $59$ |
\( (T^{3} - 9 T^{2} + \cdots + 160)^{2} \)
|
| $61$ |
\( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \)
|
| $67$ |
\( T^{6} + 132 T^{4} + \cdots + 4096 \)
|
| $71$ |
\( (T^{3} - T^{2} - 118 T - 334)^{2} \)
|
| $73$ |
\( T^{6} + 17 T^{4} + \cdots + 16 \)
|
| $79$ |
\( (T^{3} - 4 T^{2} + \cdots + 160)^{2} \)
|
| $83$ |
\( T^{6} + 328 T^{4} + \cdots + 16384 \)
|
| $89$ |
\( (T^{3} - 9 T^{2} + \cdots + 160)^{2} \)
|
| $97$ |
\( T^{6} + 89 T^{4} + \cdots + 10000 \)
|
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